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Home / Catalog / Computational Science Study of Nonlinear PDEs on Manifold Point Clouds — 4-Phase Project

Computational Science Study of Nonlinear PDEs on Manifold Point Clouds — 4-Phase Project

  • Task ID: math.manifold_pde_meshfree
  • Domain: math
  • Subdomain: numerical_odes_pde
  • Status: final
  • Benchmark set: seed42 (60 tasks)
  • Tags: meshfree, manifold, pde, laplace-beltrami, nonlinear, newton, gmls, rbf-fd, 2d, torus, convergence, robustness, sensitivity, computational-science

Runtime and requirements

  • Estimated time: 90-180 minutes
  • Python:
  • Packages: numpy, scipy
  • GPU required: no
  • Network required: yes

Public input and output contract

Inputs

  • parameters.json (data): Task parameters: R (major radius), r (minor radius), alpha (linear coeff), beta (nonlinear coeff), l (polynomial degree), N (point count).
  • points_2d.csv (data): N sample points on the torus as (theta, phi) pairs in [0, 2pi)^2.
  • h_2d.csv (data): RHS function h(theta, phi) at the N sample points.

Outputs

  • analysis.py (code): Agent's implementation script.
  • results/u_pred.csv (data): Numerical solution at the given N points (one float per line).
  • results/discretization/convergence_test.json (data): Convergence test results at 5 densities: max_error, rms_error, estimated_order.
  • results/discretization/stencil_validation.json (data): Stencil weight vectors at 5 random sample points.
  • results/discretization/discretization_report.md (data): Method description, convergence analysis, comparison with theory.
  • results/solver/solver_diagnostics.json (data): Newton solver diagnostics: residual history, step sizes, backtracking.
  • results/solver/convergence_plot_data.csv (data): Iteration, log10 residual norm, log10 step norm for plotting.
  • results/robustness/point_cloud_sensitivity.json (data): Solution change at noise levels sigma=0.01, 0.02, 0.05.
  • results/robustness/parameter_sensitivity.json (data): Solution change at (alpha,beta) perturbations +/-10%, +/-20%.
  • results/robustness/linear_vs_nonlinear.csv (data): u_linear and u_nonlinear at N points.
  • results/robustness/robustness_report.md (data): Analysis of noise robustness and parameter sensitivity.
  • results/analysis/solution_analysis.md (data): Scientific interpretation of solution structure and method limitations.
  • results/analysis/solution_features.json (data): Key solution features: max/min location, mean, std, nonlinear contribution.

Public repository files

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Provenance

This task is included in the current seed42 benchmark set at dataset revision f11a199f71fb4b854ac43a1bf548d5df519141a9.

This page is generated only from files tracked in the public ASI-Bench repository at commit f18382f03faf. Submission bundles, run logs, private scoring configuration, and private reference answers are not read by this page generator.